A solution of Braess' approximaiton problem on powers of the distance function

Abstract

The polynomial approximation behaviour of the class of functions Fs:R2(x0,y0)−>R,Fs(x,y)=((x−x0)2+(y−y0)2)(−s),sin(0,infty), is studied in [Bra01]. There it is claimed that the obtained results can be embedded in a more general setting. This conjecture will be confirmed and complemented by a different approach than in [Bra01]. The key is to connect the approximation rate of F_s with its holomorphic continuability for which the classical Bernstein approximation theorem is linked with the convexity of best approximants. Approximation results of this kind also play a vital role in the numerical treatment of elliptic differential equations [Sau].